Lemma 86.9.5. Let $S$ be a scheme. Let $X \to Y$ be a proper, flat morphism of algebraic spaces which is of finite presentation. There exists a relative dualizing complex $(\omega _{X/Y}^\bullet , \tau )$.
Proof. Choose a étale hypercovering $b : V \to Y$ such that each $V_ n = \coprod _{i \in I_ n} Y_{n, i}$ with $Y_{n, i}$ affine. This is possible by Hypercoverings, Lemma 25.12.2 and Remark 25.12.9 (to replace the hypercovering produced in the lemma by the one having disjoint unions in each degree). Denote $X_{n, i} = Y_{n, i} \times _ Y X$ and $U_ n = V_ n \times _ Y X$ so that we obtain an étale hypercovering $a : U \to X$ (Hypercoverings, Lemma 25.12.4) with $U_ n = \coprod X_{n, i}$. For each $n, i$ there exists a relative dualizing complex $(\omega _{n, i}^\bullet , \tau _{n, i})$ on $X_{n, i}/Y_{n, i}$. See discussion following Definition 86.9.1. For $\varphi : [m] \to [n]$ and $i \in I_ n$ consider the morphisms $g_{\varphi , i} : Y_{n, i} \to Y_{m, \alpha (\varphi )}$ and $g'_{\varphi , i} : X_{n, i} \to X_{m, \alpha (\varphi )}$ which are part of the structure of the given hypercoverings (Hypercoverings, Section 25.12). Then we have a unique isomorphisms
\[ \iota _{n, i, \varphi } : (L(g'_{n, i})^*\omega _{n, i}^\bullet , Lg_{n, i}^*\tau _{n, i}) \longrightarrow (\omega _{m, \alpha (\varphi )(i)}^\bullet , \tau _{m, \alpha (\varphi )(i)}) \]
of pairs, see discussion following Definition 86.9.1. Observe that $\omega _{n, i}^\bullet $ has vanishing negative self exts on $X_{n, i}$ by Lemma 86.9.2. Denote $(\omega _ n^\bullet , \tau _ n)$ the pair on $U_ n/V_ n$ constructed using the pairs $(\omega _{n, i}^\bullet , \tau _{n, i})$ for $i \in I_ n$. For $\varphi : [m] \to [n]$ and $i \in I_ n$ consider the morphisms $g_\varphi : V_ n \to V_ m$ and $g'_\varphi : U_ n \to U_ m$ which are part of the structure of the simplicial algebraic spaces $V$ and $U$. Then we have unique isomorphisms
\[ \iota _\varphi : (L(g'_\varphi )^*\omega _ n^\bullet , Lg_\varphi ^*\tau _ n) \longrightarrow (\omega _ m^\bullet , \tau _ m) \]
of pairs constructed from the isomorphisms on the pieces. The uniqueness guarantees that these isomorphisms satisfy the transitivity condition as formulated in Simplicial Spaces, Definition 85.14.1. The assumptions of Simplicial Spaces, Lemma 85.35.2 are satisfied for $a : U \to X$, the complexes $\omega _ n^\bullet $ and the isomorphisms $\iota _\varphi $1. Thus we obtain an object $\omega ^\bullet $ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ together with an isomorphism $\iota _0 : \omega ^\bullet |_{U_0} \to \omega _0^\bullet $ compatible with the two isomorphisms $\iota _{\delta ^1_0}$ and $\iota _{\delta ^1_1}$. Finally, we apply Simplicial Spaces, Lemma 85.35.1 to find a unique morphism
\[ \tau : Rf_*\omega ^\bullet \longrightarrow \mathcal{O}_ Y \]
whose restriction to $V_0$ agrees with $\tau _0$; some details omitted – compare with the end of the proof of Lemma 86.9.3 for example to see why we have the required vanishing of negative exts. By Lemma 86.9.4 the pair $(\omega ^\bullet , \tau )$ is a relative dualizing complex and the proof is complete. $\square$
[1] This lemma uses only $\omega _0^\bullet $ and the two maps $\delta _1^1, \delta _0^1 : [1] \to [0]$. The reader can skip the first few lines of the proof of the referenced lemma because here we actually are already given a simplicial system of the derived category of modules.
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